examples_1202

The blue curve represents \(y=f(x)\). The red curve represents \(y=g(x)\).

What equation expresses the function transformation?

\(g(x)= \frac{f(x)}{2}\)\(g(x)= f{\left(\frac{x}{3}\right)}\)\(g(x)= f{\left(\frac{x}{2}\right)}\)\(g(x)= \frac{f(x)}{3}\)\(g(x)= f{\left(3\cdot x\right)}\)\(g(x)=3\cdot f(x)\)\(g(x)=2\cdot f(x)\)\(g(x)= f{\left(2\cdot x\right)}\)

We see the points are transformed vertically, so we are expecting the operation outside the argument. We also remember that vertical transformations are "normal".

In this case, we see from blue to red, the points have stretched away from the horizontal axis by a factor of 2. In other words, the coordinate transformation is:

\[(a,b)\to \left(a~,~2\cdot b\right)\]

So, the correct answer is:

\[g(x)=2\cdot f(x)\]

It might help to visually check all the following:

\[g(-10) = 2\cdot f(-10)\] \[g(2) = 2\cdot f(2)\] \[g(4) = 2\cdot f(4)\] \[g(5) = 2\cdot f(5)\]

The blue curve represents \(y=f(x)\). The red curve represents \(y=g(x)\).

What equation expresses the function transformation?

\(g(x)= f{\left(\frac{x}{2}\right)}\)\(g(x)= \frac{f(x)}{3}\)\(g(x)= \frac{f(x)}{2}\)\(g(x)= f{\left(3\cdot x\right)}\)\(g(x)= f{\left(2\cdot x\right)}\)\(g(x)=3\cdot f(x)\)\(g(x)= f{\left(\frac{x}{3}\right)}\)\(g(x)=2\cdot f(x)\)

We see the points are transformed vertically, so we are expecting the operation outside the argument. We also remember that vertical transformations are "normal".

In this case, we see from blue to red, the points have shrunk towards the horizontal axis by a factor of 3. In other words, the coordinate transformation is:

\[(a,b)\to \left(a~,~\frac{b}{3}\right)\]

So, the correct answer is:

\[g(x)= \frac{f(x)}{3}\]

It might help to visually check all the following:

\[g(-9) = \frac{f(-9)}{3}\] \[g(-8) = \frac{f(-8)}{3}\] \[g(2) = \frac{f(2)}{3}\] \[g(6) = \frac{f(6)}{3}\]

The blue curve represents \(y=f(x)\). The red curve represents \(y=g(x)\).

What equation expresses the function transformation?

\(g(x)= f{\left(3\cdot x\right)}\)\(g(x)= f{\left(\frac{x}{3}\right)}\)\(g(x)= \frac{f(x)}{2}\)\(g(x)=3\cdot f(x)\)\(g(x)=2\cdot f(x)\)\(g(x)= \frac{f(x)}{3}\)\(g(x)= f{\left(2\cdot x\right)}\)\(g(x)= f{\left(\frac{x}{2}\right)}\)

We see the points are transformed vertically, so we are expecting the operation outside the argument. We also remember that vertical transformations are "normal".

In this case, we see from blue to red, the points have shrunk towards the horizontal axis by a factor of 2. In other words, the coordinate transformation is:

\[(a,b)\to \left(a~,~\frac{b}{2}\right)\]

So, the correct answer is:

\[g(x)= \frac{f(x)}{2}\]

It might help to visually check all the following:

\[g(-6) = \frac{f(-6)}{2}\] \[g(-1) = \frac{f(-1)}{2}\] \[g(0) = \frac{f(0)}{2}\] \[g(10) = \frac{f(10)}{2}\]

The blue curve represents \(y=f(x)\). The red curve represents \(y=g(x)\).

What equation expresses the function transformation?

\(g(x)= f{\left(\frac{x}{2}\right)}\)\(g(x)=3\cdot f(x)\)\(g(x)= \frac{f(x)}{3}\)\(g(x)=2\cdot f(x)\)\(g(x)= f{\left(3\cdot x\right)}\)\(g(x)= f{\left(\frac{x}{3}\right)}\)\(g(x)= f{\left(2\cdot x\right)}\)\(g(x)= \frac{f(x)}{2}\)

We see the points are transformed horizontally, so we are expecting the operation inside the argument. We also remember that horizontal transformations are "backwards".

In this case, we see from blue to red, the points have stretched away from the vertical axis by a factor of 3. In other words, the coordinate transformation is:

\[(a,b)\to \left(3\cdot a~,~b\right)\]

Unfortunately, when we express the transformation as an equation in function notation, we see the inverse operation. So, instead of multiplication, we see division. So, the correct answer is:

\[g(x)= f{\left(\frac{x}{3}\right)}\]

For example, in the graph we see \(g(6) = f(2) = -1\). We see this because the red curve has a point at \((6,-1)\) and the blue curve has a corresponding point at \((2,-1)\). It might help to visually check all the following:

\[g(-9) = f(-3)\] \[g(-6) = f(-2)\] \[g(3) = f(1)\] \[g(6) = f(2)\]

The blue curve represents \(y=f(x)\). The red curve represents \(y=g(x)\).

What equation expresses the function transformation?

\(g(x)= f{\left(3\cdot x\right)}\)\(g(x)=2\cdot f(x)\)\(g(x)=3\cdot f(x)\)\(g(x)= f{\left(\frac{x}{2}\right)}\)\(g(x)= \frac{f(x)}{3}\)\(g(x)= f{\left(2\cdot x\right)}\)\(g(x)= \frac{f(x)}{2}\)\(g(x)= f{\left(\frac{x}{3}\right)}\)

We see the points are transformed horizontally, so we are expecting the operation inside the argument. We also remember that horizontal transformations are "backwards".

In this case, we see from blue to red, the points have stretched away from the vertical axis by a factor of 3. In other words, the coordinate transformation is:

\[(a,b)\to \left(3\cdot a~,~b\right)\]

Unfortunately, when we express the transformation as an equation in function notation, we see the inverse operation. So, instead of multiplication, we see division. So, the correct answer is:

\[g(x)= f{\left(\frac{x}{3}\right)}\]

For example, in the graph we see \(g(9) = f(3) = 10\). We see this because the red curve has a point at \((9,10)\) and the blue curve has a corresponding point at \((3,10)\). It might help to visually check all the following:

\[g(-9) = f(-3)\] \[g(-6) = f(-2)\] \[g(6) = f(2)\] \[g(9) = f(3)\]

The blue curve represents \(y=f(x)\). The red curve represents \(y=g(x)\).

What equation expresses the function transformation?

\(g(x)= f{\left(3\cdot x\right)}\)\(g(x)= f{\left(\frac{x}{3}\right)}\)\(g(x)= \frac{f(x)}{2}\)\(g(x)=3\cdot f(x)\)\(g(x)= f{\left(2\cdot x\right)}\)\(g(x)=2\cdot f(x)\)\(g(x)= \frac{f(x)}{3}\)\(g(x)= f{\left(\frac{x}{2}\right)}\)

We see the points are transformed horizontally, so we are expecting the operation inside the argument. We also remember that horizontal transformations are "backwards".

In this case, we see from blue to red, the points have shrunk towards the vertical axis by a factor of 3. In other words, the coordinate transformation is:

\[(a,b)\to \left(\frac{a}{3}~,~b\right)\]

Unfortunately, when we express the transformation as an equation in function notation, we see the inverse operation. So, instead of division, we see multiplication. So, the correct answer is:

\[g(x)= f{\left(3\cdot x\right)}\]

For example, in the graph we see \(g(3) = f(9) = -1\). We see this because the red curve has a point at \((3,-1)\) and the blue curve has a corresponding point at \((9,-1)\). It might help to visually check all the following:

\[g(-2) = f(-6)\] \[g(0) = f(0)\] \[g(2) = f(6)\] \[g(3) = f(9)\]

The blue curve represents \(y=f(x)\). The red curve represents \(y=g(x)\).

What equation expresses the function transformation?

\(g(x)= f{\left(\frac{x}{2}\right)}\)\(g(x)= f{\left(3\cdot x\right)}\)\(g(x)=3\cdot f(x)\)\(g(x)= \frac{f(x)}{3}\)\(g(x)= f{\left(2\cdot x\right)}\)\(g(x)= f{\left(\frac{x}{3}\right)}\)\(g(x)= \frac{f(x)}{2}\)\(g(x)=2\cdot f(x)\)

We see the points are transformed vertically, so we are expecting the operation outside the argument. We also remember that vertical transformations are "normal".

In this case, we see from blue to red, the points have stretched away from the horizontal axis by a factor of 2. In other words, the coordinate transformation is:

\[(a,b)\to \left(a~,~2\cdot b\right)\]

So, the correct answer is:

\[g(x)=2\cdot f(x)\]

It might help to visually check all the following:

\[g(-3) = 2\cdot f(-3)\] \[g(3) = 2\cdot f(3)\] \[g(8) = 2\cdot f(8)\] \[g(10) = 2\cdot f(10)\]

The blue curve represents \(y=f(x)\). The red curve represents \(y=g(x)\).

What equation expresses the function transformation?

\(g(x)=3\cdot f(x)\)\(g(x)= \frac{f(x)}{3}\)\(g(x)= \frac{f(x)}{2}\)\(g(x)= f{\left(3\cdot x\right)}\)\(g(x)= f{\left(\frac{x}{2}\right)}\)\(g(x)= f{\left(\frac{x}{3}\right)}\)\(g(x)=2\cdot f(x)\)\(g(x)= f{\left(2\cdot x\right)}\)

We see the points are transformed horizontally, so we are expecting the operation inside the argument. We also remember that horizontal transformations are "backwards".

In this case, we see from blue to red, the points have shrunk towards the vertical axis by a factor of 3. In other words, the coordinate transformation is:

\[(a,b)\to \left(\frac{a}{3}~,~b\right)\]

Unfortunately, when we express the transformation as an equation in function notation, we see the inverse operation. So, instead of division, we see multiplication. So, the correct answer is:

\[g(x)= f{\left(3\cdot x\right)}\]

For example, in the graph we see \(g(1) = f(3) = 9\). We see this because the red curve has a point at \((1,9)\) and the blue curve has a corresponding point at \((3,9)\). It might help to visually check all the following:

\[g(-3) = f(-9)\] \[g(-1) = f(-3)\] \[g(0) = f(0)\] \[g(1) = f(3)\]

The blue curve represents \(y=f(x)\). The red curve represents \(y=g(x)\).

What equation expresses the function transformation?

\(g(x)=2\cdot f(x)\)\(g(x)= f{\left(\frac{x}{2}\right)}\)\(g(x)= f{\left(2\cdot x\right)}\)\(g(x)= f{\left(\frac{x}{3}\right)}\)\(g(x)= \frac{f(x)}{2}\)\(g(x)=3\cdot f(x)\)\(g(x)= f{\left(3\cdot x\right)}\)\(g(x)= \frac{f(x)}{3}\)

We see the points are transformed horizontally, so we are expecting the operation inside the argument. We also remember that horizontal transformations are "backwards".

In this case, we see from blue to red, the points have shrunk towards the vertical axis by a factor of 3. In other words, the coordinate transformation is:

\[(a,b)\to \left(\frac{a}{3}~,~b\right)\]

Unfortunately, when we express the transformation as an equation in function notation, we see the inverse operation. So, instead of division, we see multiplication. So, the correct answer is:

\[g(x)= f{\left(3\cdot x\right)}\]

For example, in the graph we see \(g(2) = f(6) = 0\). We see this because the red curve has a point at \((2,0)\) and the blue curve has a corresponding point at \((6,0)\). It might help to visually check all the following:

\[g(-3) = f(-9)\] \[g(0) = f(0)\] \[g(1) = f(3)\] \[g(2) = f(6)\]

The blue curve represents \(y=f(x)\). The red curve represents \(y=g(x)\).

What equation expresses the function transformation?

\(g(x)= \frac{f(x)}{3}\)\(g(x)=3\cdot f(x)\)\(g(x)= f{\left(\frac{x}{2}\right)}\)\(g(x)= f{\left(\frac{x}{3}\right)}\)\(g(x)= \frac{f(x)}{2}\)\(g(x)=2\cdot f(x)\)\(g(x)= f{\left(2\cdot x\right)}\)\(g(x)= f{\left(3\cdot x\right)}\)

We see the points are transformed vertically, so we are expecting the operation outside the argument. We also remember that vertical transformations are "normal".

In this case, we see from blue to red, the points have shrunk towards the horizontal axis by a factor of 3. In other words, the coordinate transformation is:

\[(a,b)\to \left(a~,~\frac{b}{3}\right)\]

So, the correct answer is:

\[g(x)= \frac{f(x)}{3}\]

It might help to visually check all the following:

\[g(-2) = \frac{f(-2)}{3}\] \[g(-1) = \frac{f(-1)}{3}\] \[g(3) = \frac{f(3)}{3}\] \[g(8) = \frac{f(8)}{3}\]

Consider the curve represented by the equation below:

\[y~=~\frac{1}{x+6}-4\]

Does the curve \(y=\frac{1}{x+6}-4\) have a horizontal asymptote?

Yes, at x=-6.Yes, at x=-4.Yes, at x=4.Yes, at x=6.Yes, at y=-6.Yes, at y=-4.Yes, at y=4.Yes, at y=6.No.

Does the curve \(y=\frac{1}{x+6}-4\) have a vertical asymptote?

Yes, at x=-6.Yes, at x=-4.Yes, at x=4.Yes, at x=6.Yes, at y=-6.Yes, at y=-4.Yes, at y=4.Yes, at y=6.No.

The horizontal and vertical asymptotes are shown below as a red dashed lines. The blue curve represents \(y=\frac{1}{x+6}-4\).

Notice, the parent function, \(y=\frac{1}{x}\), has a horizontal asymptote at \(y=0\) and a vertical asymptote at \(x=0\). When we shift the points up or down, the horizontal asymptote also shifts up or down. When we shift the points left or right, the vertical asymptote also shifts left or right.

Consider the curve represented by the equation below:

\[y~=~\frac{1}{x-5}-6\]

Does the curve \(y=\frac{1}{x-5}-6\) have a horizontal asymptote?

Yes, at x=-6.Yes, at x=-5.Yes, at x=5.Yes, at x=6.Yes, at y=-6.Yes, at y=-5.Yes, at y=5.Yes, at y=6.No.

Does the curve \(y=\frac{1}{x-5}-6\) have a vertical asymptote?

Yes, at x=-6.Yes, at x=-5.Yes, at x=5.Yes, at x=6.Yes, at y=-6.Yes, at y=-5.Yes, at y=5.Yes, at y=6.No.

The horizontal and vertical asymptotes are shown below as a red dashed lines. The blue curve represents \(y=\frac{1}{x-5}-6\).

Notice, the parent function, \(y=\frac{1}{x}\), has a horizontal asymptote at \(y=0\) and a vertical asymptote at \(x=0\). When we shift the points up or down, the horizontal asymptote also shifts up or down. When we shift the points left or right, the vertical asymptote also shifts left or right.

Consider the curve represented by the equation below:

\[y~=~\log_2(x-5)+6\]

Does the curve \(y=\log_2(x-5)+6\) have a horizontal asymptote?

Yes, at x=-6.Yes, at x=-5.Yes, at x=5.Yes, at x=6.Yes, at y=-6.Yes, at y=-5.Yes, at y=5.Yes, at y=6.No.

Does the curve \(y=\log_2(x-5)+6\) have a vertical asymptote?

Yes, at x=-6.Yes, at x=-5.Yes, at x=5.Yes, at x=6.Yes, at y=-6.Yes, at y=-5.Yes, at y=5.Yes, at y=6.No.

The vertical asymptote is shown below as a red dashed line. The blue curve represents \(y=\log_2(x-5)+6\).

Notice, the parent function, \(y=\log_2(x)\), has a vertical asymptote at \(x=0\). When we shift the points left or right, the vertical asymptote also shifts left or right.

Also, kind of strange, but a vertical line is indicated by \(x\) equaling a constant.

Consider the curve represented by the equation below:

\[y~=~\frac{1}{x+6}-5\]

Does the curve \(y=\frac{1}{x+6}-5\) have a horizontal asymptote?

Yes, at x=-6.Yes, at x=-5.Yes, at x=5.Yes, at x=6.Yes, at y=-6.Yes, at y=-5.Yes, at y=5.Yes, at y=6.No.

Does the curve \(y=\frac{1}{x+6}-5\) have a vertical asymptote?

Yes, at x=-6.Yes, at x=-5.Yes, at x=5.Yes, at x=6.Yes, at y=-6.Yes, at y=-5.Yes, at y=5.Yes, at y=6.No.

The horizontal and vertical asymptotes are shown below as a red dashed lines. The blue curve represents \(y=\frac{1}{x+6}-5\).

Notice, the parent function, \(y=\frac{1}{x}\), has a horizontal asymptote at \(y=0\) and a vertical asymptote at \(x=0\). When we shift the points up or down, the horizontal asymptote also shifts up or down. When we shift the points left or right, the vertical asymptote also shifts left or right.

Consider the curve represented by the equation below:

\[y~=~\log_2(x+1)-5\]

Does the curve \(y=\log_2(x+1)-5\) have a horizontal asymptote?

Yes, at x=-5.Yes, at x=-1.Yes, at x=1.Yes, at x=5.Yes, at y=-5.Yes, at y=-1.Yes, at y=1.Yes, at y=5.No.

Does the curve \(y=\log_2(x+1)-5\) have a vertical asymptote?

Yes, at x=-5.Yes, at x=-1.Yes, at x=1.Yes, at x=5.Yes, at y=-5.Yes, at y=-1.Yes, at y=1.Yes, at y=5.No.

The vertical asymptote is shown below as a red dashed line. The blue curve represents \(y=\log_2(x+1)-5\).

Notice, the parent function, \(y=\log_2(x)\), has a vertical asymptote at \(x=0\). When we shift the points left or right, the vertical asymptote also shifts left or right.

Also, kind of strange, but a vertical line is indicated by \(x\) equaling a constant.

Consider the curve represented by the equation below:

\[y~=~\frac{1}{x+2}+5\]

Does the curve \(y=\frac{1}{x+2}+5\) have a horizontal asymptote?

Yes, at x=-5.Yes, at x=-2.Yes, at x=2.Yes, at x=5.Yes, at y=-5.Yes, at y=-2.Yes, at y=2.Yes, at y=5.No.

Does the curve \(y=\frac{1}{x+2}+5\) have a vertical asymptote?

Yes, at x=-5.Yes, at x=-2.Yes, at x=2.Yes, at x=5.Yes, at y=-5.Yes, at y=-2.Yes, at y=2.Yes, at y=5.No.

The horizontal and vertical asymptotes are shown below as a red dashed lines. The blue curve represents \(y=\frac{1}{x+2}+5\).

Notice, the parent function, \(y=\frac{1}{x}\), has a horizontal asymptote at \(y=0\) and a vertical asymptote at \(x=0\). When we shift the points up or down, the horizontal asymptote also shifts up or down. When we shift the points left or right, the vertical asymptote also shifts left or right.

Consider the curve represented by the equation below:

\[y~=~2^{x+6}-2\]

Does the curve \(y=2^{x+6}-2\) have a horizontal asymptote?

Yes, at x=-6.Yes, at x=-2.Yes, at x=2.Yes, at x=6.Yes, at y=-6.Yes, at y=-2.Yes, at y=2.Yes, at y=6.No.

Does the curve \(y=2^{x+6}-2\) have a vertical asymptote?

Yes, at x=-6.Yes, at x=-2.Yes, at x=2.Yes, at x=6.Yes, at y=-6.Yes, at y=-2.Yes, at y=2.Yes, at y=6.No.

The horizontal asymptote is shown below as a red dashed line. The blue curve represents \(y=2^{x+6}-2\).

Notice, the parent function, \(y=2^x\), has a horizontal asymptote at \(y=0\). When we shift the points up or down, the horizontal asymptote also shifts up or down.

Also, kind of strange, but a horizontal line is indicated by \(y\) equaling a constant.

Consider the curve represented by the equation below:

\[y~=~\log_2(x-6)-5\]

Does the curve \(y=\log_2(x-6)-5\) have a horizontal asymptote?

Yes, at x=-6.Yes, at x=-5.Yes, at x=5.Yes, at x=6.Yes, at y=-6.Yes, at y=-5.Yes, at y=5.Yes, at y=6.No.

Does the curve \(y=\log_2(x-6)-5\) have a vertical asymptote?

Yes, at x=-6.Yes, at x=-5.Yes, at x=5.Yes, at x=6.Yes, at y=-6.Yes, at y=-5.Yes, at y=5.Yes, at y=6.No.

The vertical asymptote is shown below as a red dashed line. The blue curve represents \(y=\log_2(x-6)-5\).

Notice, the parent function, \(y=\log_2(x)\), has a vertical asymptote at \(x=0\). When we shift the points left or right, the vertical asymptote also shifts left or right.

Also, kind of strange, but a vertical line is indicated by \(x\) equaling a constant.

Consider the curve represented by the equation below:

\[y~=~\log_2(x+3)-6\]

Does the curve \(y=\log_2(x+3)-6\) have a horizontal asymptote?

Yes, at x=-6.Yes, at x=-3.Yes, at x=3.Yes, at x=6.Yes, at y=-6.Yes, at y=-3.Yes, at y=3.Yes, at y=6.No.

Does the curve \(y=\log_2(x+3)-6\) have a vertical asymptote?

Yes, at x=-6.Yes, at x=-3.Yes, at x=3.Yes, at x=6.Yes, at y=-6.Yes, at y=-3.Yes, at y=3.Yes, at y=6.No.

The vertical asymptote is shown below as a red dashed line. The blue curve represents \(y=\log_2(x+3)-6\).

Notice, the parent function, \(y=\log_2(x)\), has a vertical asymptote at \(x=0\). When we shift the points left or right, the vertical asymptote also shifts left or right.

Also, kind of strange, but a vertical line is indicated by \(x\) equaling a constant.

Consider the curve represented by the equation below:

\[y~=~\frac{1}{x+6}+1\]

Does the curve \(y=\frac{1}{x+6}+1\) have a horizontal asymptote?

Yes, at x=-6.Yes, at x=-1.Yes, at x=1.Yes, at x=6.Yes, at y=-6.Yes, at y=-1.Yes, at y=1.Yes, at y=6.No.

Does the curve \(y=\frac{1}{x+6}+1\) have a vertical asymptote?

Yes, at x=-6.Yes, at x=-1.Yes, at x=1.Yes, at x=6.Yes, at y=-6.Yes, at y=-1.Yes, at y=1.Yes, at y=6.No.

The horizontal and vertical asymptotes are shown below as a red dashed lines. The blue curve represents \(y=\frac{1}{x+6}+1\).

Notice, the parent function, \(y=\frac{1}{x}\), has a horizontal asymptote at \(y=0\) and a vertical asymptote at \(x=0\). When we shift the points up or down, the horizontal asymptote also shifts up or down. When we shift the points left or right, the vertical asymptote also shifts left or right.

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-1)-4y = f(x-1)+4y = f(x-4)-1y = f(x-4)+1y = f(x+1)-4y = f(x+1)+4y = f(x+4)-1y = f(x+4)+1

The equation used to produce the plot was: \[y~=~\sqrt[3]{x-1}+4\]

In other words, the parent function was the cube root function: \(f(x)=\sqrt[3]{x}\). The parent function was shifted 1 units right and 4 units up, so the function-transformation equation was \(y = f(x-1)+4\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-2)-5y = f(x-2)+5y = f(x-5)-2y = f(x-5)+2y = f(x+2)-5y = f(x+2)+5y = f(x+5)-2y = f(x+5)+2

The equation used to produce the plot was: \[y~=~\log_{2}(x+2)+5\]

In other words, the parent function was the logarithmic function: \(f(x)=\log_{2}(x)\). The parent function was shifted 2 units left and 5 units up, so the function-transformation equation was \(y = f(x+2)+5\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-1)-6y = f(x-1)+6y = f(x-6)-1y = f(x-6)+1y = f(x+1)-6y = f(x+1)+6y = f(x+6)-1y = f(x+6)+1

The equation used to produce the plot was: \[y~=~\frac{1}{x+1}+6\]

In other words, the parent function was the reciprocal function: \(f(x)=\frac{1}{x}\). The parent function was shifted 1 units left and 6 units up, so the function-transformation equation was \(y = f(x+1)+6\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-1)-6y = f(x-1)+6y = f(x-6)-1y = f(x-6)+1y = f(x+1)-6y = f(x+1)+6y = f(x+6)-1y = f(x+6)+1

The equation used to produce the plot was: \[y~=~2^{x+6}-1\]

In other words, the parent function was the exponential function: \(f(x)=2^{x}\). The parent function was shifted 6 units left and 1 units down, so the function-transformation equation was \(y = f(x+6)-1\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-5)-6y = f(x-5)+6y = f(x-6)-5y = f(x-6)+5y = f(x+5)-6y = f(x+5)+6y = f(x+6)-5y = f(x+6)+5

The equation used to produce the plot was: \[y~=~\frac{1}{x+6}-5\]

In other words, the parent function was the reciprocal function: \(f(x)=\frac{1}{x}\). The parent function was shifted 6 units left and 5 units down, so the function-transformation equation was \(y = f(x+6)-5\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-1)-5y = f(x-1)+5y = f(x-5)-1y = f(x-5)+1y = f(x+1)-5y = f(x+1)+5y = f(x+5)-1y = f(x+5)+1

The equation used to produce the plot was: \[y~=~\lvert x-5 \rvert+1\]

In other words, the parent function was the absolute value function: \(f(x)=\lvert x \rvert\). The parent function was shifted 5 units right and 1 units up, so the function-transformation equation was \(y = f(x-5)+1\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-5)-6y = f(x-5)+6y = f(x-6)-5y = f(x-6)+5y = f(x+5)-6y = f(x+5)+6y = f(x+6)-5y = f(x+6)+5

The equation used to produce the plot was: \[y~=~(x-6)^2+5\]

In other words, the parent function was the quadratic function: \(f(x)=x^2\). The parent function was shifted 6 units right and 5 units up, so the function-transformation equation was \(y = f(x-6)+5\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-5)-6y = f(x-5)+6y = f(x-6)-5y = f(x-6)+5y = f(x+5)-6y = f(x+5)+6y = f(x+6)-5y = f(x+6)+5

The equation used to produce the plot was: \[y~=~2^{x-5}-6\]

In other words, the parent function was the exponential function: \(f(x)=2^{x}\). The parent function was shifted 5 units right and 6 units down, so the function-transformation equation was \(y = f(x-5)-6\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-1)-4y = f(x-1)+4y = f(x-4)-1y = f(x-4)+1y = f(x+1)-4y = f(x+1)+4y = f(x+4)-1y = f(x+4)+1

The equation used to produce the plot was: \[y~=~2^{x-1}+4\]

In other words, the parent function was the exponential function: \(f(x)=2^{x}\). The parent function was shifted 1 units right and 4 units up, so the function-transformation equation was \(y = f(x-1)+4\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-1)-4y = f(x-1)+4y = f(x-4)-1y = f(x-4)+1y = f(x+1)-4y = f(x+1)+4y = f(x+4)-1y = f(x+4)+1

The equation used to produce the plot was: \[y~=~\frac{1}{x-4}-1\]

In other words, the parent function was the reciprocal function: \(f(x)=\frac{1}{x}\). The parent function was shifted 4 units right and 1 units down, so the function-transformation equation was \(y = f(x-4)-1\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-3)-5y = f(x-3)+5y = f(x-5)-3y = f(x-5)+3y = f(x+3)-5y = f(x+3)+5y = f(x+5)-3y = f(x+5)+3

The equation used to produce the plot was: \[y~=~\sqrt{x+3}+5\]

In other words, the parent function was the square root function: \(f(x)=\sqrt{x}\). The parent function was shifted 3 units left and 5 units up, so the function-transformation equation was \(y = f(x+3)+5\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-2)-5y = f(x-2)+5y = f(x-5)-2y = f(x-5)+2y = f(x+2)-5y = f(x+2)+5y = f(x+5)-2y = f(x+5)+2

The equation used to produce the plot was: \[y~=~\lvert x+2 \rvert-5\]

In other words, the parent function was the absolute value function: \(f(x)=\lvert x \rvert\). The parent function was shifted 2 units left and 5 units down, so the function-transformation equation was \(y = f(x+2)-5\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-2)-4y = f(x-2)+4y = f(x-4)-2y = f(x-4)+2y = f(x+2)-4y = f(x+2)+4y = f(x+4)-2y = f(x+4)+2

The equation used to produce the plot was: \[y~=~2^{x+2}-4\]

In other words, the parent function was the exponential function: \(f(x)=2^{x}\). The parent function was shifted 2 units left and 4 units down, so the function-transformation equation was \(y = f(x+2)-4\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-4)-5y = f(x-4)+5y = f(x-5)-4y = f(x-5)+4y = f(x+4)-5y = f(x+4)+5y = f(x+5)-4y = f(x+5)+4

The equation used to produce the plot was: \[y~=~(x+5)^2-4\]

In other words, the parent function was the quadratic function: \(f(x)=x^2\). The parent function was shifted 5 units left and 4 units down, so the function-transformation equation was \(y = f(x+5)-4\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-1)-4y = f(x-1)+4y = f(x-4)-1y = f(x-4)+1y = f(x+1)-4y = f(x+1)+4y = f(x+4)-1y = f(x+4)+1

The equation used to produce the plot was: \[y~=~\sqrt{x-4}+1\]

In other words, the parent function was the square root function: \(f(x)=\sqrt{x}\). The parent function was shifted 4 units right and 1 units up, so the function-transformation equation was \(y = f(x-4)+1\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-1)-6y = f(x-1)+6y = f(x-6)-1y = f(x-6)+1y = f(x+1)-6y = f(x+1)+6y = f(x+6)-1y = f(x+6)+1

The equation used to produce the plot was: \[y~=~(x+1)^3+6\]

In other words, the parent function was the cubic function: \(f(x)=x^3\). The parent function was shifted 1 units left and 6 units up, so the function-transformation equation was \(y = f(x+1)+6\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-1)-2y = f(x-1)+2y = f(x-2)-1y = f(x-2)+1y = f(x+1)-2y = f(x+1)+2y = f(x+2)-1y = f(x+2)+1

The equation used to produce the plot was: \[y~=~(x-2)^2-1\]

In other words, the parent function was the quadratic function: \(f(x)=x^2\). The parent function was shifted 2 units right and 1 units down, so the function-transformation equation was \(y = f(x-2)-1\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-1)-5y = f(x-1)+5y = f(x-5)-1y = f(x-5)+1y = f(x+1)-5y = f(x+1)+5y = f(x+5)-1y = f(x+5)+1

The equation used to produce the plot was: \[y~=~\sqrt[3]{x+5}+1\]

In other words, the parent function was the cube root function: \(f(x)=\sqrt[3]{x}\). The parent function was shifted 5 units left and 1 units up, so the function-transformation equation was \(y = f(x+5)+1\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-2)-3y = f(x-2)+3y = f(x-3)-2y = f(x-3)+2y = f(x+2)-3y = f(x+2)+3y = f(x+3)-2y = f(x+3)+2

The equation used to produce the plot was: \[y~=~(x+2)^3-3\]

In other words, the parent function was the cubic function: \(f(x)=x^3\). The parent function was shifted 2 units left and 3 units down, so the function-transformation equation was \(y = f(x+2)-3\).

Here is a graph of the parent (in grey) and child (in blue).

Below are some common parent functions.

names	expressions
absolute value	\(f(x)=\lvert x \rvert\)
quadratic	\(f(x)=x^2\)
cubic	\(f(x)=x^3\)
reciprocal	\(f(x)=\frac{1}{x}\)
square root	\(f(x)=\sqrt{x}\)
cube root	\(f(x)=\sqrt[3]{x}\)
exponential	\(f(x)=2^{x}\)
logarithmic	\(f(x)=\log_{2}(x)\)

One of those parent functions has been translated (shifted) to produce the graph below:

• Which parent function was used?

absolute valuequadraticcubicreciprocalsquare rootcube rootexponentiallogarithmic

• What translation was used?

y = f(x-1)-4y = f(x-1)+4y = f(x-4)-1y = f(x-4)+1y = f(x+1)-4y = f(x+1)+4y = f(x+4)-1y = f(x+4)+1

The equation used to produce the plot was: \[y~=~(x+4)^2-1\]

In other words, the parent function was the quadratic function: \(f(x)=x^2\). The parent function was shifted 4 units left and 1 units down, so the function-transformation equation was \(y = f(x+4)-1\).

Here is a graph of the parent (in grey) and child (in blue).

Consider the shifted square-root function graphed below:

Which equation represents the plotted curve?

\(y=\sqrt{x-3}-5\)\(y=\sqrt{x-3}+5\)\(y=\sqrt{x-5}-3\)\(y=\sqrt{x-5}+3\)\(y=\sqrt{x+3}-5\)\(y=\sqrt{x+3}+5\)\(y=\sqrt{x+5}-3\)\(y=\sqrt{x+5}+3\)

The parent function, \(y=\sqrt{x}\), has been shifted \(3\) units left and \(5\) units down.

The correct answer is \[y=\sqrt{x+3}-5\]

Notice the horizontal translation is counterintuitive. The horizontal coordinate transformation is the INVERSE of the argument expression, whereas the vertical coordinate transformation is simply the outer operations.

\[(a,b)~\to~(a-3,b-5)\]

where \((a,b)\) is a solution to the parent equation \(\sqrt{a}=b\). For example, \(\sqrt{0}=0\) or \(\sqrt{1}=1\) or \(\sqrt{4}=2\) or \(\sqrt{9}=3\)... Thus from curve \(y=\sqrt{x}\) to curve \(y=\sqrt{x+3}-5\), we have the following coordinate transformations:

\[(0,0) \to (-3,-5)\] \[(1,1) \to (-2,-4)\] \[(4,2) \to (1,-3)\] \[(9,3) \to (6,-2)\]

Consider the shifted square-root function graphed below:

Which equation represents the plotted curve?

\(y=\sqrt{x-3}-5\)\(y=\sqrt{x-3}+5\)\(y=\sqrt{x-5}-3\)\(y=\sqrt{x-5}+3\)\(y=\sqrt{x+3}-5\)\(y=\sqrt{x+3}+5\)\(y=\sqrt{x+5}-3\)\(y=\sqrt{x+5}+3\)

The parent function, \(y=\sqrt{x}\), has been shifted \(3\) units left and \(5\) units up.

The correct answer is \[y=\sqrt{x+3}+5\]

Notice the horizontal translation is counterintuitive. The horizontal coordinate transformation is the INVERSE of the argument expression, whereas the vertical coordinate transformation is simply the outer operations.

\[(a,b)~\to~(a-3,b+5)\]

where \((a,b)\) is a solution to the parent equation \(\sqrt{a}=b\). For example, \(\sqrt{0}=0\) or \(\sqrt{1}=1\) or \(\sqrt{4}=2\) or \(\sqrt{9}=3\)... Thus from curve \(y=\sqrt{x}\) to curve \(y=\sqrt{x+3}+5\), we have the following coordinate transformations:

\[(0,0) \to (-3,5)\] \[(1,1) \to (-2,6)\] \[(4,2) \to (1,7)\] \[(9,3) \to (6,8)\]

Consider the shifted square-root function graphed below:

Which equation represents the plotted curve?

\(y=\sqrt{x-2}-6\)\(y=\sqrt{x-2}+6\)\(y=\sqrt{x-6}-2\)\(y=\sqrt{x-6}+2\)\(y=\sqrt{x+2}-6\)\(y=\sqrt{x+2}+6\)\(y=\sqrt{x+6}-2\)\(y=\sqrt{x+6}+2\)

The parent function, \(y=\sqrt{x}\), has been shifted \(6\) units right and \(2\) units up.

The correct answer is \[y=\sqrt{x-6}+2\]

Notice the horizontal translation is counterintuitive. The horizontal coordinate transformation is the INVERSE of the argument expression, whereas the vertical coordinate transformation is simply the outer operations.

\[(a,b)~\to~(a+6,b+2)\]

where \((a,b)\) is a solution to the parent equation \(\sqrt{a}=b\). For example, \(\sqrt{0}=0\) or \(\sqrt{1}=1\) or \(\sqrt{4}=2\) or \(\sqrt{9}=3\)... Thus from curve \(y=\sqrt{x}\) to curve \(y=\sqrt{x-6}+2\), we have the following coordinate transformations:

\[(0,0) \to (6,2)\] \[(1,1) \to (7,3)\] \[(4,2) \to (10,4)\] \[(9,3) \to (15,5)\]

Consider the shifted square-root function graphed below:

Which equation represents the plotted curve?

\(y=\sqrt{x-2}-4\)\(y=\sqrt{x-2}+4\)\(y=\sqrt{x-4}-2\)\(y=\sqrt{x-4}+2\)\(y=\sqrt{x+2}-4\)\(y=\sqrt{x+2}+4\)\(y=\sqrt{x+4}-2\)\(y=\sqrt{x+4}+2\)

The parent function, \(y=\sqrt{x}\), has been shifted \(2\) units left and \(4\) units up.

The correct answer is \[y=\sqrt{x+2}+4\]

Notice the horizontal translation is counterintuitive. The horizontal coordinate transformation is the INVERSE of the argument expression, whereas the vertical coordinate transformation is simply the outer operations.

\[(a,b)~\to~(a-2,b+4)\]

where \((a,b)\) is a solution to the parent equation \(\sqrt{a}=b\). For example, \(\sqrt{0}=0\) or \(\sqrt{1}=1\) or \(\sqrt{4}=2\) or \(\sqrt{9}=3\)... Thus from curve \(y=\sqrt{x}\) to curve \(y=\sqrt{x+2}+4\), we have the following coordinate transformations:

\[(0,0) \to (-2,4)\] \[(1,1) \to (-1,5)\] \[(4,2) \to (2,6)\] \[(9,3) \to (7,7)\]

Consider the shifted square-root function graphed below:

Which equation represents the plotted curve?

\(y=\sqrt{x-2}-5\)\(y=\sqrt{x-2}+5\)\(y=\sqrt{x-5}-2\)\(y=\sqrt{x-5}+2\)\(y=\sqrt{x+2}-5\)\(y=\sqrt{x+2}+5\)\(y=\sqrt{x+5}-2\)\(y=\sqrt{x+5}+2\)

The parent function, \(y=\sqrt{x}\), has been shifted \(5\) units right and \(2\) units down.

The correct answer is \[y=\sqrt{x-5}-2\]

Notice the horizontal translation is counterintuitive. The horizontal coordinate transformation is the INVERSE of the argument expression, whereas the vertical coordinate transformation is simply the outer operations.

\[(a,b)~\to~(a+5,b-2)\]

where \((a,b)\) is a solution to the parent equation \(\sqrt{a}=b\). For example, \(\sqrt{0}=0\) or \(\sqrt{1}=1\) or \(\sqrt{4}=2\) or \(\sqrt{9}=3\)... Thus from curve \(y=\sqrt{x}\) to curve \(y=\sqrt{x-5}-2\), we have the following coordinate transformations:

\[(0,0) \to (5,-2)\] \[(1,1) \to (6,-1)\] \[(4,2) \to (9,0)\] \[(9,3) \to (14,1)\]

Consider the shifted square-root function graphed below:

Which equation represents the plotted curve?

\(y=\sqrt{x-2}-3\)\(y=\sqrt{x-2}+3\)\(y=\sqrt{x-3}-2\)\(y=\sqrt{x-3}+2\)\(y=\sqrt{x+2}-3\)\(y=\sqrt{x+2}+3\)\(y=\sqrt{x+3}-2\)\(y=\sqrt{x+3}+2\)

The parent function, \(y=\sqrt{x}\), has been shifted \(3\) units right and \(2\) units down.

The correct answer is \[y=\sqrt{x-3}-2\]

Notice the horizontal translation is counterintuitive. The horizontal coordinate transformation is the INVERSE of the argument expression, whereas the vertical coordinate transformation is simply the outer operations.

\[(a,b)~\to~(a+3,b-2)\]

where \((a,b)\) is a solution to the parent equation \(\sqrt{a}=b\). For example, \(\sqrt{0}=0\) or \(\sqrt{1}=1\) or \(\sqrt{4}=2\) or \(\sqrt{9}=3\)... Thus from curve \(y=\sqrt{x}\) to curve \(y=\sqrt{x-3}-2\), we have the following coordinate transformations:

\[(0,0) \to (3,-2)\] \[(1,1) \to (4,-1)\] \[(4,2) \to (7,0)\] \[(9,3) \to (12,1)\]

Consider the shifted square-root function graphed below:

Which equation represents the plotted curve?

\(y=\sqrt{x-1}-4\)\(y=\sqrt{x-1}+4\)\(y=\sqrt{x-4}-1\)\(y=\sqrt{x-4}+1\)\(y=\sqrt{x+1}-4\)\(y=\sqrt{x+1}+4\)\(y=\sqrt{x+4}-1\)\(y=\sqrt{x+4}+1\)

The parent function, \(y=\sqrt{x}\), has been shifted \(4\) units right and \(1\) units up.

The correct answer is \[y=\sqrt{x-4}+1\]

Notice the horizontal translation is counterintuitive. The horizontal coordinate transformation is the INVERSE of the argument expression, whereas the vertical coordinate transformation is simply the outer operations.

\[(a,b)~\to~(a+4,b+1)\]

where \((a,b)\) is a solution to the parent equation \(\sqrt{a}=b\). For example, \(\sqrt{0}=0\) or \(\sqrt{1}=1\) or \(\sqrt{4}=2\) or \(\sqrt{9}=3\)... Thus from curve \(y=\sqrt{x}\) to curve \(y=\sqrt{x-4}+1\), we have the following coordinate transformations:

\[(0,0) \to (4,1)\] \[(1,1) \to (5,2)\] \[(4,2) \to (8,3)\] \[(9,3) \to (13,4)\]

Consider the shifted square-root function graphed below:

Which equation represents the plotted curve?

\(y=\sqrt{x-1}-2\)\(y=\sqrt{x-1}+2\)\(y=\sqrt{x-2}-1\)\(y=\sqrt{x-2}+1\)\(y=\sqrt{x+1}-2\)\(y=\sqrt{x+1}+2\)\(y=\sqrt{x+2}-1\)\(y=\sqrt{x+2}+1\)

The parent function, \(y=\sqrt{x}\), has been shifted \(2\) units left and \(1\) units down.

The correct answer is \[y=\sqrt{x+2}-1\]

Notice the horizontal translation is counterintuitive. The horizontal coordinate transformation is the INVERSE of the argument expression, whereas the vertical coordinate transformation is simply the outer operations.

\[(a,b)~\to~(a-2,b-1)\]

where \((a,b)\) is a solution to the parent equation \(\sqrt{a}=b\). For example, \(\sqrt{0}=0\) or \(\sqrt{1}=1\) or \(\sqrt{4}=2\) or \(\sqrt{9}=3\)... Thus from curve \(y=\sqrt{x}\) to curve \(y=\sqrt{x+2}-1\), we have the following coordinate transformations:

\[(0,0) \to (-2,-1)\] \[(1,1) \to (-1,0)\] \[(4,2) \to (2,1)\] \[(9,3) \to (7,2)\]

Consider the shifted square-root function graphed below:

Which equation represents the plotted curve?

\(y=\sqrt{x-1}-4\)\(y=\sqrt{x-1}+4\)\(y=\sqrt{x-4}-1\)\(y=\sqrt{x-4}+1\)\(y=\sqrt{x+1}-4\)\(y=\sqrt{x+1}+4\)\(y=\sqrt{x+4}-1\)\(y=\sqrt{x+4}+1\)

The parent function, \(y=\sqrt{x}\), has been shifted \(1\) units left and \(4\) units down.

The correct answer is \[y=\sqrt{x+1}-4\]

Notice the horizontal translation is counterintuitive. The horizontal coordinate transformation is the INVERSE of the argument expression, whereas the vertical coordinate transformation is simply the outer operations.

\[(a,b)~\to~(a-1,b-4)\]

where \((a,b)\) is a solution to the parent equation \(\sqrt{a}=b\). For example, \(\sqrt{0}=0\) or \(\sqrt{1}=1\) or \(\sqrt{4}=2\) or \(\sqrt{9}=3\)... Thus from curve \(y=\sqrt{x}\) to curve \(y=\sqrt{x+1}-4\), we have the following coordinate transformations:

\[(0,0) \to (-1,-4)\] \[(1,1) \to (0,-3)\] \[(4,2) \to (3,-2)\] \[(9,3) \to (8,-1)\]

Consider the shifted square-root function graphed below:

Which equation represents the plotted curve?

\(y=\sqrt{x-2}-6\)\(y=\sqrt{x-2}+6\)\(y=\sqrt{x-6}-2\)\(y=\sqrt{x-6}+2\)\(y=\sqrt{x+2}-6\)\(y=\sqrt{x+2}+6\)\(y=\sqrt{x+6}-2\)\(y=\sqrt{x+6}+2\)

The parent function, \(y=\sqrt{x}\), has been shifted \(6\) units left and \(2\) units down.

The correct answer is \[y=\sqrt{x+6}-2\]

Notice the horizontal translation is counterintuitive. The horizontal coordinate transformation is the INVERSE of the argument expression, whereas the vertical coordinate transformation is simply the outer operations.

\[(a,b)~\to~(a-6,b-2)\]

where \((a,b)\) is a solution to the parent equation \(\sqrt{a}=b\). For example, \(\sqrt{0}=0\) or \(\sqrt{1}=1\) or \(\sqrt{4}=2\) or \(\sqrt{9}=3\)... Thus from curve \(y=\sqrt{x}\) to curve \(y=\sqrt{x+6}-2\), we have the following coordinate transformations:

\[(0,0) \to (-6,-2)\] \[(1,1) \to (-5,-1)\] \[(4,2) \to (-2,0)\] \[(9,3) \to (3,1)\]

If \((x+n)\) is a factor of the polynomial below, and \(n\) is an integer, then what is the value of \(n\)?

\[{5}x^{2}+{14}x-{3}\]

\[{5}x^{2}+{14}x-{3}\]

You'll want to factor the expression.

Set up a box with what you know... Notice that the leading (quadratic) coefficient, \(a=5\), is prime. This allows us to be more confident in how this expression factors.

You can get this by guessing and checking. To make a smart guess, find a pair of integers that multiply to get \(-3\), and use those...

Eventually, because \((-1)\cdot(3)=-3\), you will guess the following factored form:

\[({5}x-{1})\cdot(x+{3})\]

And you will check it with a box to see that it works.

\[{5}x^{2}+{15}x-x-{3}\]

Combine like terms.

\[{5}x^{2}+{14}x-{3}\]

So it worked! (Because it matches the given standard-form expression.) Thus, in factored form, the expression is:

\[({5}x-{1})\cdot(x+{3})\]

The problem asked for the constant of the second factor, because it asked for the constant of the factor with a linear coefficient of 1.

\[n=3\]

If \((x+n)\) is a factor of the polynomial below, and \(n\) is an integer, then what is the value of \(n\)?

\[{7}x^{2}-{25}x-{12}\]

\[{7}x^{2}-{25}x-{12}\]

You'll want to factor the expression.

Set up a box with what you know... Notice that the leading (quadratic) coefficient, \(a=7\), is prime. This allows us to be more confident in how this expression factors.

You can get this by guessing and checking. To make a smart guess, find a pair of integers that multiply to get \(-12\), and use those...

Eventually, because \((3)\cdot(-4)=-12\), you will guess the following factored form:

\[({7}x+{3})\cdot(x-{4})\]

And you will check it with a box to see that it works.

\[{7}x^{2}-{28}x+{3}x-{12}\]

Combine like terms.

\[{7}x^{2}-{25}x-{12}\]

So it worked! (Because it matches the given standard-form expression.) Thus, in factored form, the expression is:

\[({7}x+{3})\cdot(x-{4})\]

The problem asked for the constant of the second factor, because it asked for the constant of the factor with a linear coefficient of 1.

\[n=-4\]

If \((x+n)\) is a factor of the polynomial below, and \(n\) is an integer, then what is the value of \(n\)?

\[{7}x^{2}+{3}x-{4}\]

\[{7}x^{2}+{3}x-{4}\]

You'll want to factor the expression.

Set up a box with what you know... Notice that the leading (quadratic) coefficient, \(a=7\), is prime. This allows us to be more confident in how this expression factors.

You can get this by guessing and checking. To make a smart guess, find a pair of integers that multiply to get \(-4\), and use those...

Eventually, because \((-4)\cdot(1)=-4\), you will guess the following factored form:

\[({7}x-{4})\cdot(x+{1})\]

And you will check it with a box to see that it works.

\[{7}x^{2}+{7}x-{4}x-{4}\]

Combine like terms.

\[{7}x^{2}+{3}x-{4}\]

So it worked! (Because it matches the given standard-form expression.) Thus, in factored form, the expression is:

\[({7}x-{4})\cdot(x+{1})\]

The problem asked for the constant of the second factor, because it asked for the constant of the factor with a linear coefficient of 1.

\[n=1\]

If \((x+n)\) is a factor of the polynomial below, and \(n\) is an integer, then what is the value of \(n\)?

\[{3}x^{2}-{26}x+{48}\]

\[{3}x^{2}-{26}x+{48}\]

You'll want to factor the expression.

Set up a box with what you know... Notice that the leading (quadratic) coefficient, \(a=3\), is prime. This allows us to be more confident in how this expression factors.

You can get this by guessing and checking. To make a smart guess, find a pair of integers that multiply to get \(48\), and use those...

Eventually, because \((-8)\cdot(-6)=48\), you will guess the following factored form:

\[({3}x-{8})\cdot(x-{6})\]

And you will check it with a box to see that it works.

\[{3}x^{2}-{18}x-{8}x+{48}\]

Combine like terms.

\[{3}x^{2}-{26}x+{48}\]

So it worked! (Because it matches the given standard-form expression.) Thus, in factored form, the expression is:

\[({3}x-{8})\cdot(x-{6})\]

The problem asked for the constant of the second factor, because it asked for the constant of the factor with a linear coefficient of 1.

\[n=-6\]

If \((x+n)\) is a factor of the polynomial below, and \(n\) is an integer, then what is the value of \(n\)?

\[{3}x^{2}+x-{24}\]

\[{3}x^{2}+x-{24}\]

You'll want to factor the expression.

Set up a box with what you know... Notice that the leading (quadratic) coefficient, \(a=3\), is prime. This allows us to be more confident in how this expression factors.

You can get this by guessing and checking. To make a smart guess, find a pair of integers that multiply to get \(-24\), and use those...

Eventually, because \((-8)\cdot(3)=-24\), you will guess the following factored form:

\[({3}x-{8})\cdot(x+{3})\]

And you will check it with a box to see that it works.

\[{3}x^{2}+{9}x-{8}x-{24}\]

Combine like terms.

\[{3}x^{2}+x-{24}\]

So it worked! (Because it matches the given standard-form expression.) Thus, in factored form, the expression is:

\[({3}x-{8})\cdot(x+{3})\]

The problem asked for the constant of the second factor, because it asked for the constant of the factor with a linear coefficient of 1.

\[n=3\]

If \((x+n)\) is a factor of the polynomial below, and \(n\) is an integer, then what is the value of \(n\)?

\[{3}x^{2}-{25}x+{50}\]

\[{3}x^{2}-{25}x+{50}\]

You'll want to factor the expression.

Set up a box with what you know... Notice that the leading (quadratic) coefficient, \(a=3\), is prime. This allows us to be more confident in how this expression factors.

You can get this by guessing and checking. To make a smart guess, find a pair of integers that multiply to get \(50\), and use those...

Eventually, because \((-10)\cdot(-5)=50\), you will guess the following factored form:

\[({3}x-{10})\cdot(x-{5})\]

And you will check it with a box to see that it works.

\[{3}x^{2}-{15}x-{10}x+{50}\]

Combine like terms.

\[{3}x^{2}-{25}x+{50}\]

So it worked! (Because it matches the given standard-form expression.) Thus, in factored form, the expression is:

\[({3}x-{10})\cdot(x-{5})\]

The problem asked for the constant of the second factor, because it asked for the constant of the factor with a linear coefficient of 1.

\[n=-5\]

If \((x+n)\) is a factor of the polynomial below, and \(n\) is an integer, then what is the value of \(n\)?

\[{7}x^{2}-{60}x-{27}\]

\[{7}x^{2}-{60}x-{27}\]

You'll want to factor the expression.

Set up a box with what you know... Notice that the leading (quadratic) coefficient, \(a=7\), is prime. This allows us to be more confident in how this expression factors.

You can get this by guessing and checking. To make a smart guess, find a pair of integers that multiply to get \(-27\), and use those...

Eventually, because \((3)\cdot(-9)=-27\), you will guess the following factored form:

\[({7}x+{3})\cdot(x-{9})\]

And you will check it with a box to see that it works.

\[{7}x^{2}-{63}x+{3}x-{27}\]

Combine like terms.

\[{7}x^{2}-{60}x-{27}\]

So it worked! (Because it matches the given standard-form expression.) Thus, in factored form, the expression is:

\[({7}x+{3})\cdot(x-{9})\]

The problem asked for the constant of the second factor, because it asked for the constant of the factor with a linear coefficient of 1.

\[n=-9\]

If \((x+n)\) is a factor of the polynomial below, and \(n\) is an integer, then what is the value of \(n\)?

\[{3}x^{2}+{10}x+{3}\]

\[{3}x^{2}+{10}x+{3}\]

You'll want to factor the expression.

Set up a box with what you know... Notice that the leading (quadratic) coefficient, \(a=3\), is prime. This allows us to be more confident in how this expression factors.

You can get this by guessing and checking. To make a smart guess, find a pair of integers that multiply to get \(3\), and use those...

Eventually, because \((1)\cdot(3)=3\), you will guess the following factored form:

\[({3}x+{1})\cdot(x+{3})\]

And you will check it with a box to see that it works.

\[{3}x^{2}+{9}x+x+{3}\]

Combine like terms.

\[{3}x^{2}+{10}x+{3}\]

So it worked! (Because it matches the given standard-form expression.) Thus, in factored form, the expression is:

\[({3}x+{1})\cdot(x+{3})\]

The problem asked for the constant of the second factor, because it asked for the constant of the factor with a linear coefficient of 1.

\[n=3\]

If \((x+n)\) is a factor of the polynomial below, and \(n\) is an integer, then what is the value of \(n\)?

\[{5}x^{2}-{44}x+{32}\]

\[{5}x^{2}-{44}x+{32}\]

You'll want to factor the expression.

Set up a box with what you know... Notice that the leading (quadratic) coefficient, \(a=5\), is prime. This allows us to be more confident in how this expression factors.

You can get this by guessing and checking. To make a smart guess, find a pair of integers that multiply to get \(32\), and use those...

Eventually, because \((-4)\cdot(-8)=32\), you will guess the following factored form:

\[({5}x-{4})\cdot(x-{8})\]

And you will check it with a box to see that it works.

\[{5}x^{2}-{40}x-{4}x+{32}\]

Combine like terms.

\[{5}x^{2}-{44}x+{32}\]

So it worked! (Because it matches the given standard-form expression.) Thus, in factored form, the expression is:

\[({5}x-{4})\cdot(x-{8})\]

The problem asked for the constant of the second factor, because it asked for the constant of the factor with a linear coefficient of 1.

\[n=-8\]

If \((x+n)\) is a factor of the polynomial below, and \(n\) is an integer, then what is the value of \(n\)?

\[{7}x^{2}+{10}x-{8}\]

\[{7}x^{2}+{10}x-{8}\]

You'll want to factor the expression.

Set up a box with what you know... Notice that the leading (quadratic) coefficient, \(a=7\), is prime. This allows us to be more confident in how this expression factors.

You can get this by guessing and checking. To make a smart guess, find a pair of integers that multiply to get \(-8\), and use those...

Eventually, because \((-4)\cdot(2)=-8\), you will guess the following factored form:

\[({7}x-{4})\cdot(x+{2})\]

And you will check it with a box to see that it works.

\[{7}x^{2}+{14}x-{4}x-{8}\]

Combine like terms.

\[{7}x^{2}+{10}x-{8}\]

So it worked! (Because it matches the given standard-form expression.) Thus, in factored form, the expression is:

\[({7}x-{4})\cdot(x+{2})\]

The problem asked for the constant of the second factor, because it asked for the constant of the factor with a linear coefficient of 1.

\[n=2\]

If \(8x^2-648\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[8x^2-648\]

Notice that 8 and 648 are both multiples of 8. Factor out this constant.

\[8(x^2-81)\]

In the parentheses is a difference of squares. Thus, we just need to find the square root of 81, and then it is easy to find the factors.

\[8(x+9)(x-9)\]

Thus, \[a=8\] \[p=9\] \[q=9\]

If \(6x^2-486\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[6x^2-486\]

Notice that 6 and 486 are both multiples of 6. Factor out this constant.

\[6(x^2-81)\]

In the parentheses is a difference of squares. Thus, we just need to find the square root of 81, and then it is easy to find the factors.

\[6(x+9)(x-9)\]

Thus, \[a=6\] \[p=9\] \[q=9\]

If \(7x^2-567\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[7x^2-567\]

Notice that 7 and 567 are both multiples of 7. Factor out this constant.

\[7(x^2-81)\]

In the parentheses is a difference of squares. Thus, we just need to find the square root of 81, and then it is easy to find the factors.

\[7(x+9)(x-9)\]

Thus, \[a=7\] \[p=9\] \[q=9\]

If \(7x^2-700\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[7x^2-700\]

Notice that 7 and 700 are both multiples of 7. Factor out this constant.

\[7(x^2-100)\]

In the parentheses is a difference of squares. Thus, we just need to find the square root of 100, and then it is easy to find the factors.

\[7(x+10)(x-10)\]

Thus, \[a=7\] \[p=10\] \[q=10\]

If \(6x^2-600\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[6x^2-600\]

Notice that 6 and 600 are both multiples of 6. Factor out this constant.

\[6(x^2-100)\]

In the parentheses is a difference of squares. Thus, we just need to find the square root of 100, and then it is easy to find the factors.

\[6(x+10)(x-10)\]

Thus, \[a=6\] \[p=10\] \[q=10\]

If \(8x^2-392\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[8x^2-392\]

Notice that 8 and 392 are both multiples of 8. Factor out this constant.

\[8(x^2-49)\]

In the parentheses is a difference of squares. Thus, we just need to find the square root of 49, and then it is easy to find the factors.

\[8(x+7)(x-7)\]

Thus, \[a=8\] \[p=7\] \[q=7\]

If \(4x^2-16\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[4x^2-16\]

Notice that 4 and 16 are both multiples of 4. Factor out this constant.

\[4(x^2-4)\]

In the parentheses is a difference of squares. Thus, we just need to find the square root of 4, and then it is easy to find the factors.

\[4(x+2)(x-2)\]

Thus, \[a=4\] \[p=2\] \[q=2\]

If \(9x^2-576\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[9x^2-576\]

Notice that 9 and 576 are both multiples of 9. Factor out this constant.

\[9(x^2-64)\]

In the parentheses is a difference of squares. Thus, we just need to find the square root of 64, and then it is easy to find the factors.

\[9(x+8)(x-8)\]

Thus, \[a=9\] \[p=8\] \[q=8\]

If \(10x^2-250\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[10x^2-250\]

Notice that 10 and 250 are both multiples of 10. Factor out this constant.

\[10(x^2-25)\]

In the parentheses is a difference of squares. Thus, we just need to find the square root of 25, and then it is easy to find the factors.

\[10(x+5)(x-5)\]

Thus, \[a=10\] \[p=5\] \[q=5\]

If \(2x^2-200\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[2x^2-200\]

Notice that 2 and 200 are both multiples of 2. Factor out this constant.

\[2(x^2-100)\]

In the parentheses is a difference of squares. Thus, we just need to find the square root of 100, and then it is easy to find the factors.

\[2(x+10)(x-10)\]

Thus, \[a=2\] \[p=10\] \[q=10\]

If \({8}x^{2}-{40}x-{288}\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[{8}x^{2}-{40}x-{288}\]

Notice that 8 and -40 and -288 are all multiples of 8. Factor out this constant.

\[8(x^{2}-{5}x-{36})\]

In the parentheses is a quadratic trinomial with an implied leading coefficient of 1. Thus, we simply need two numbers whose sum is -5 and whose product is -36. This puzzle is solved with 4 and -9.

\[8(x+4)(x-9)\]

Thus, \[a=8\] \[p=4\] \[q=9\]

If \({9}x^{2}-{18}x-{216}\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[{9}x^{2}-{18}x-{216}\]

Notice that 9 and -18 and -216 are all multiples of 9. Factor out this constant.

\[9(x^{2}-{2}x-{24})\]

In the parentheses is a quadratic trinomial with an implied leading coefficient of 1. Thus, we simply need two numbers whose sum is -2 and whose product is -24. This puzzle is solved with 4 and -6.

\[9(x+4)(x-6)\]

Thus, \[a=9\] \[p=4\] \[q=6\]

If \({3}x^{2}+{3}x-{216}\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[{3}x^{2}+{3}x-{216}\]

Notice that 3 and 3 and -216 are all multiples of 3. Factor out this constant.

\[3(x^{2}+x-{72})\]

In the parentheses is a quadratic trinomial with an implied leading coefficient of 1. Thus, we simply need two numbers whose sum is 1 and whose product is -72. This puzzle is solved with 9 and -8.

\[3(x+9)(x-8)\]

Thus, \[a=3\] \[p=9\] \[q=8\]

If \({5}x^{2}-{5}x-{280}\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[{5}x^{2}-{5}x-{280}\]

Notice that 5 and -5 and -280 are all multiples of 5. Factor out this constant.

\[5(x^{2}-x-{56})\]

In the parentheses is a quadratic trinomial with an implied leading coefficient of 1. Thus, we simply need two numbers whose sum is -1 and whose product is -56. This puzzle is solved with 7 and -8.

\[5(x+7)(x-8)\]

Thus, \[a=5\] \[p=7\] \[q=8\]

If \({5}x^{2}-{25}x-{250}\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[{5}x^{2}-{25}x-{250}\]

Notice that 5 and -25 and -250 are all multiples of 5. Factor out this constant.

\[5(x^{2}-{5}x-{50})\]

In the parentheses is a quadratic trinomial with an implied leading coefficient of 1. Thus, we simply need two numbers whose sum is -5 and whose product is -50. This puzzle is solved with 5 and -10.

\[5(x+5)(x-10)\]

Thus, \[a=5\] \[p=5\] \[q=10\]

If \({10}x^{2}+{60}x-{400}\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[{10}x^{2}+{60}x-{400}\]

Notice that 10 and 60 and -400 are all multiples of 10. Factor out this constant.

\[10(x^{2}+{6}x-{40})\]

In the parentheses is a quadratic trinomial with an implied leading coefficient of 1. Thus, we simply need two numbers whose sum is 6 and whose product is -40. This puzzle is solved with 10 and -4.

\[10(x+10)(x-4)\]

Thus, \[a=10\] \[p=10\] \[q=4\]

If \({4}x^{2}-{12}x-{40}\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[{4}x^{2}-{12}x-{40}\]

Notice that 4 and -12 and -40 are all multiples of 4. Factor out this constant.

\[4(x^{2}-{3}x-{10})\]

In the parentheses is a quadratic trinomial with an implied leading coefficient of 1. Thus, we simply need two numbers whose sum is -3 and whose product is -10. This puzzle is solved with 2 and -5.

\[4(x+2)(x-5)\]

Thus, \[a=4\] \[p=2\] \[q=5\]

If \({5}x^{2}-{5}x-{60}\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[{5}x^{2}-{5}x-{60}\]

Notice that 5 and -5 and -60 are all multiples of 5. Factor out this constant.

\[5(x^{2}-x-{12})\]

In the parentheses is a quadratic trinomial with an implied leading coefficient of 1. Thus, we simply need two numbers whose sum is -1 and whose product is -12. This puzzle is solved with 3 and -4.

\[5(x+3)(x-4)\]

Thus, \[a=5\] \[p=3\] \[q=4\]

If \({3}x^{2}+{18}x-{120}\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[{3}x^{2}+{18}x-{120}\]

Notice that 3 and 18 and -120 are all multiples of 3. Factor out this constant.

\[3(x^{2}+{6}x-{40})\]

In the parentheses is a quadratic trinomial with an implied leading coefficient of 1. Thus, we simply need two numbers whose sum is 6 and whose product is -40. This puzzle is solved with 10 and -4.

\[3(x+10)(x-4)\]

Thus, \[a=3\] \[p=10\] \[q=4\]

If \({2}x^{2}+{14}x-{36}\) is fully factored, it can be expressed in the form \(a(x+p)(x-q)\), where \(a\), \(p\), and \(q\) are positive integers.

Find \(a\), \(p\), and \(q\).

• \(a=\) 2345678910
• \(p=\) 2345678910
• \(q=\) 2345678910

\[{2}x^{2}+{14}x-{36}\]

Notice that 2 and 14 and -36 are all multiples of 2. Factor out this constant.

\[2(x^{2}+{7}x-{18})\]

In the parentheses is a quadratic trinomial with an implied leading coefficient of 1. Thus, we simply need two numbers whose sum is 7 and whose product is -18. This puzzle is solved with 9 and -2.

\[2(x+9)(x-2)\]

Thus, \[a=2\] \[p=9\] \[q=2\]